1. Field of the Invention
The subject invention relates generally to a method of generating Fresnel reflectors utilized in specular image display systems that are used to magnify the appearance of planar objects for viewing by the eye.
2. Description Relative to the Prior Art
Concave reflective mirrors are sometimes used in a variety of imaging systems. It has long been known that a concave reflective mirror surface of revolution can be described by a conic equation of the general form: EQU y.sup.2 -2rx+px.sup.2 =0,
where y is the perpendicular distance from the axis of revolution to any given point on the surface, x is the distance from the point to a plane passing through the (y=0) origin and perpendicular to the axis of revolution, r is rms value of x and y, and p is a constant.
An alternative equation which can be shown is equivalent to the above equation is discussed in the user's manual to the Super Oslo optical computation program developed by Sinclair Optics of Pittsford, New York. This equation describes the sag of the curve as the distance measured parallel to the axis of revolution to any point on the surface from that point on the surface which coincides with the axis of revolution. ##EQU1## where z is the sag of that point as described in the preceding sentence, r is the rms value of any given point (x,y) on the surface, and c, e, e, f, g, are constants.
Since the optical surfaces most frequently employed by optical designers are sections of spheres, this case was assumed by the software designer to be the configuration of choice and is the default configuration unless aspherizing is specifically asked for. The manual describes the sphere as a special case of this equation where c=1/r=is constant for all points on the surface.
This optical design program also allows the designer to define the surface of a Fresnel lens. The manual defines this as "a hypothetical surface that has the power of a curved surface, but which is actually flat. In Super Oslo, a Fresnel surface is implemented by treating the surface as flat for the purpose of ray transfer, but curved for the purpose of refraction." (Reflection is a special case of refraction where the index of refraction is taken to be -1.0)
The method of surface determination in Super Oslo and other optical computing programs is primarily one of choosing: 1) what optical parameters of a system can be varied, 2) the limits of said variation, 3) the functions of system performance to improve, and 4) weighting factors for the variables and merit function which in effect establish preferences. The allowable parameter variables are changed slightly, the system error functions are calculated, and a number of iterations of this process repeated until the error functions have become minimized. As discussed by Dr. Robert Hopkins, in an article in Optical Engineering Design, (December 1988/Vol. 27, No. 12 pp. 1019-1026) the process that early lens designers used was limited by practicality to paraxial, meridional, and Coddington's rays and accumulated hands-on experience when designing lenses since calculations were made manually with arithmetic references to logarithm tables. Many of the error functions developed are based on the hypothesis the a perfect optical system will direct all the rays from a given object point to a single image point and attempts to reach this goal were evaluated by optical star testing in the shop. The evolution of optical design has primarily advanced along the line of increased computing efficiency which allows the designer to evaluate higher order aberrations in almost real time. However, the design approach remains essentially the same. Consequently, the process is still most efficient when dealing with rotationally symmetric elements, and centered, or nearly centered optical systems.
A typical prior art problem is to design an optical display which magnifies the appearance of an object. This is illustrated in FIG. 1 where the object subtends an angle .PHI..sub.o and through the effect of the display system appears to subtend an angle .PHI..
U.S. Pat. No. 4,717,248 describes a video display device where a spherical reflector is used in the off-axis specular display configuration to magnify a video screen. A plano, second mirror is also used to reinvert the image. It is suggested that tilting adjustments can be made to the video tube to eliminate trapezoidal distortion in the image. However, the ray-tracing technique herein described shows that a uniformly spaced square grid pattern on the video tube would be subject to non-linear stretching in the orthogonal directions parallel to the vertical and horizontal video image axis which cannot be eliminated by tilting the object plane. This stretching becomes a practical limitation to the potential magnifying power which could otherwise be increased by reducing the radius of curvature or relocating the object closer to the conjugate focus of the eye position. An improved surface to eliminate this pyramidal error, might be conceptually visualized, but would be difficult for the design process algorithms used in optical design programs like Super Oslo. The nominal object off-axis condition is further complicated by the fact that an observer's eyes are also off-axis in the + and - directions orthogonal to the decentering of the video tube.
The properties of a conic-section mirror can be approximated by a mirror utilizing the principles of a Fresnel lens. For example, such a construction is disclosed in my commonly assigned copending U.S. patent application Ser. No. 559,026, entitled REFLECTIVE IMAGE DISPLAY, filed simultaneously herewith in the name of Roy Y. Taylor and is hereby incorporated by reference. The modern method of fabrication is to turn inverse facets on a master mold plate using a special numerically-controlled lathe and a tool containing the inverse facet curvature. The slope of the tool is changed as a function of the radius from the axis of revolution of the mold plate, thus assuring the proper slope geometry and smoothness. Fresnel lenses can subsequently be replicated from the master in relatively thin sections without sinks or distortions of profile by compression molding. It would therefore be a potential advantage to replace the spherical mirror of U.S. Pat. No. 4,717,248 with a Fresnel mirror which was constructed to retroreflect like a spherical mirror. However, if a configuration similar to U.S. Pat. No. 4,717,248 is used with a Fresnel spherical mirror, a likewise image stretching distortion error results that leads to a likewise limitation of the invention's usefulness.
However, numerically-controlled tool guidance can also produce more complex surfaces than rotationally-symmetric surfaces. If the tool is programmed to swing through an angle about an axis perpendicular to the axis of revolution which is a function of the degree of revolution from some reference point on the workpiece, a Fresnel equivalent to an off-axis ellipse can be generated. This type of Fresnel surface was described in an article in the Jan. 1982 issue of APPLIED OPTICS (Vol. 21, No. 2), "Unusual Optics of the Polaroid SX-70 Land Camera", by William T. Plummer, "Although the focus screen is spread flat and its grooves are circular, astigmatism has been introduced to it by wobbling the diamond cutting tool two cycles per revolution, so that the relationship between slope and distance from the optical center has been arranged differently along the longitudinal and lateral axes." This article further elaborate on how the equation for this curve was determined: "The concave mirror is described by a eighth-order polynomial of revolution with the axis just below the part." The purpose of this concave mirror was described in the U.S. Pat. No. 4,006,971 by William Plummer and a similar mirror in U.S. Pat. No. 3,690,240 by Nathan Gold, assignors to Polaroid Corporation, was to reflect light emanating from a real or apparent point to be reflected therefrom and imaged at a predetermined remote exit pupil. Although not so stated, this suggests that the error function used to optimize the surface parameters was the point spread function at the desired remote exit pupil location.